A Kill Quest is a special type of quest awarded by a Kill Collector. It involves killing a certain type of creature (specified by the kill collector in his dialog). Extra experience points are awarded per kill up to a maximum number of kills (which is almost always 60 kills). After completing the quest (by killing at least the max number of creatures, and then talking to that kill collector), the words "YOU'VE RECEIVED AN INFLUENCE AWARD!" appear in the middle of the screen. A kill quest reward may be a title, a trophy, a tactics fragment, or (most often) nothing.
For example, The Finest Thread is the Empire Chapter 1 kill quest.